The key point here is the omission of $\mathbb{Q}^n_+$ from $S$. The two convex hulls $P$ and $Q$ provide a twist to the argument, and to begin just consider $S_p=\mathbb{Z}^n_+\setminus(mP+\mathbb{Q}^n_+)$.

$\mathbb{Q}^n_+$ can be $\{0,\dots,0\}$ and other variations, and so integer points of $P$ create infinite cubes from $P+\mathbb{Q}^n_+$. We can call the collection of infinite cubes the **shadow** of $P$. As $|S|$ is finite, $P$ must contain a point on each axis. We can also see that $mP$ in this case is contained in the shadow of $P$, and so we need only need to consider $\mathbb{Z}^n_+\setminus(P+Q+\mathbb{Q}^n_+)$

Adding $Q$ back into the definition of $S$ means we need to consider the minimal integer coordinates of $P$ and $Q$. Each integer point in $P$ and/or $Q$ gives rise to a shadow, and these points are eliminated from $S$. The number of integer points in the remaining non-shadowed space (a sort of jig-jagged 'cube') gives $|S|$.

If we work through your example:

$P=conv((5,0,0),(0,3,0),(0,0,11),(4,2,7))$

$Q=conv((7,0,0),(0,9,0),(0,0,4),(3,8,2))$

then the following lines bound $S$.

In this case we then have $|S|\le4.2.3=24$

These come from the minimum value for each axis e.g. for the x-axis $(7,0,0)$ is in the shadow of $(5,0,0)$.

We then need to count integer points inside the convex hulls in the shadow of other given coordinates and that those inside or on the convex hull, and subtract these to get a final value for $|S|$, which needs explicit examples.